The two-body problem is the n-body problem for two bodies of masses 
under mutual Newtonian gravitational attraction.
Their center of mass moves with constant velocity,
while their relative position vector 
satisfies
 |
where 
is the gravitational constant (Heggie 2006).
If the specific angular momentum 
is nonzero, the motion lies in a plane perpendicular to 
. In polar coordinates,
with the angular origin chosen along the direction of closest approach,
 |
where ![p=||h||^2/[G(m_1+m_2)]](/images/equations/Two-BodyProblem/Inline6.svg)
and 
is the eccentricity. The trajectory is an ellipse
for 
,
a parabola for 
, and a hyperbola for 
. The case 
is a circle; zero angular momentum
instead gives radial motion.
By Newton's shell theorem, the same exterior interaction applies to two nonoverlapping spherically symmetric bodies.
-Body Problem (Classical)." In Encyclopedia of Mathematical
Physics (Ed. J.-P. Françoise, G. L. Naber, and S. T. Tsou).
Oxford, England: Elsevier, pp. 575-582, 2006.